Exercise 3, § 2

Assume that $k$ is of characteristic 0. Let $\mathfrak{g}$ be a semi-simple Lie algebra. Let E be the set of commutative subalgebras of $\mathfrak{g}$ all of whose elements are semi-simple in $\mathfrak{g}$. Then the Cartan subalgebras of $\mathfrak{g}$ are the maximal elements of E. (Use Th. 2 and Prop. 10.)

In particular, the union of the Cartan subalgebras of $\mathfrak{g}$ is equal to the set of semi-simple elements of $\mathfrak{g}$.